01 - Combine cubed watermelon and lime juice in a blender. Puree until completely smooth, approximately 30-45 seconds.
02 - Pour the puree through a fine-mesh sieve into a pitcher, pressing with a spoon to extract maximum liquid. Discard any remaining pulp.
03 - Divide the strained watermelon puree evenly among 4 tall glasses, filling each about one-quarter full.
04 - Place one generous scoop of vanilla ice cream into the center of each glass.
05 - Slowly pour the chilled lemon-lime soda over the ice cream and puree, pouring down the side of the glass to control foaming.
06 - Top each float with fresh mint leaves and watermelon balls or slices as desired.
07 - Serve immediately with a straw and long spoon for stirring and scooping.